Charged Knots

  Jonathan Simon in Ideal Knots (1998):

• "Suppose you have a knotted loop on a string, and you spread an electric charge along the string and then let go; what will happen?"
• This question has been a common "cocktail party" topic among knot theorists for many years, ...

  Consider a Charged Trefoil

  Prime charged knots ... are tight

  Composite charged knots ... are factored

  Model & Simulations:

• Results obtained from off-lattice Monte Carlo simulations of N monomers,
  • each a sphere of diameter b;
  • connected by tethers such that 

  Energy & lengthscales:

• Coulomb interaction, with screening length :

• Effect of chaning the Bjerrum length   

  Screened Interactions:

•  In most physically relevant situations, Coulomb interactions are highly screened;
•  ~1nm for DNA in vivo, but larger values can be achieved in vitro.

Do tight knots disappear for finite screening lengths?

Electrostatic (Odijk-Skolnick-Fixman) persistence length:    

• The tight knot is the global energy minimum as long as    is comparable to size of the open chain.
• The tight knot is a local energy minimum as long as    is larger than the bare persistence length.

  Dynamics:

• Tight knots (metastable) slow down the relaxational dynamics:

• The higher the charge density, the slower is the relaxation to the final state.
• A knot in an open chain is untied by diffusion

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